Does 0.99999 recurring equal 1?

Question

Thank you all for your responses.

Keep them coming. Strange how I'm a dingbat and yet this has caused an excellent topic for debate?

Answer 1

Will the people who continue to answer "No" to this question please take a class and learn some maths?!?!?

Yes, 0.999999... recurring, repeating forever, DOES equal 1. It's merely another form of writing 1, such as 3/3.

What all do you need for proof? Here's an easy one:

1/3 = 0.333333... repeating.
2/3 = 0.666666... repeating.
Add the two lines and you get
3/3 = 0.999999... repeating. 3/3 = 1, so 0.99999... repeating = 1.

Not formal enough for a proof? Here is algebraic one:

Let x = 0.99999... repeating.
Multiply both sides by ten.
10x = 9.99999... repeating. [On the right, multiplying by 10 is a simple move of the decimal point.]
Now find the difference of these two lines by subtracting the top from the bottom.
10x - x = 9.99999... - 0.99999...
9x = 9
x = 1
If x = 0.99999... repeating (given) and x = 1 (solved), then
0.99999... repeating = 1.

How about infinite sums?

0.99999... repeating = 9(0.1)¹ + 9(0.1)² + 9(0.1)³ + ...
This is an infinite sum of ∑ 9· (0.1)^k, where k goes from 1 to ∞.
∑ = a / (1 - r), where a is the first term in the series and r is the common ratio of successive terms.
0.99999... = ∑ = 0.9 / (1 - 0.1) = 0.9 / 0.9 = 1.

People who can't believe 0.99999... repeating = 1 are the same people to whom Zeno wrote over 2400 years ago, believing Achilles will never catch the tortoise in the race. http://en.wikipedia.org/wiki/Zeno's_paradoxes#Achilles_and_the_tortoise

[Edited to add comment]
No, you're not a dingbat for asking a question. The dingbats are the ones answering incorrectly! ;-P

Answer 2

Okay so there are some heated opinions on both sides of the debate, but it's a very good problem to think through yourself.

What I want you to focus on is, what does 0.99999 recurring *mean*? Or any decimal expansion, like 3.1415... or whatever.

A number doesn't change. It's not 0.9 one moment and then 0.99 the next, as if some outside person is changing it by contemplating it. It's stuck somewhere on the number line, no matter how you describe it.

And if I ask, what is the difference between 1 and 0.99999 recurring, it's going to have to be a fixed number too, not some "infinitely small" thing. There has to be an answer; you can always subtract numbers. And it's easy to see, as many have pointed out, that the difference, if positive, would have to be smaller than any other positive number. If not positive, it must be zero.

Can it be that there is a positive number smaller than any other positive number? No, then it would be smaller than itself (or half itself). Contradiction.

Do you believe that 1/2 + 1/4 + 1/8 + 1/16 + ...=1? (Classic Achilles & the Hare problem.)
That's the same as saying 0.11111 recurring in binary is equal to 1. Your problem is the same idea base ten.

Anyway think it through yourself. The world can use people who can look at both sides of a debate and make an intelligent decision.

Best wishes

Answer 3

At first I said No but I was considering a finite number of decimal places.

The stumbling block here is whether or not the decimal terminates.

If the decimal terminates i.e. is finite then 0.99999 to however many decimals it terminates at is not equal to 1.

But if the decimal doesn't terminate and the 9's go on forever (which is what this question is about) then 0.999 recurring does equal 1

To find a number on the Real number line that comes between 0.999 recurring and 1 you would need 0.999 recurring to terminate which it doesn't.

Answer 4

Despite what your intuition might be telling you, 0.99999.... and 1 are exactly the same thing!

It's related to what's called the "density of real numbers." That means that, between any two distinct (i.e., different) real numbers, there's another number between them. (It's easy to find it, too -- just average the two numbers.)

So, I challenge you to find a number between 0.9999... and 1. Go ahead and look for it. It's not there! :)

Part of the difficulty is that, when we see the number 0.9999... written down, we read from left to right -- an action that takes time to perform -- and so we tend to think of those 9's as being "appended" or "added" to the number, one by one. (Or maybe I should say "nine by nine." ;-) ) But that's a problem of reading and notation: all the 9's are part of the number, because a number's value doesn't change. That's the problem with talking about "limits" in this context: a single number doesn't have a limit. It is what it is. And 0.9999..., in its instant, not-approaching-anything-ness, is equal to 1 because there are no numbers between them.

If that's not enough to convince you, here's an algebraic "proof:"

Let x = 0.9999....
Multiplying both sides by 10:
10x = 9.9999....
Subtracting the first equation from the second:
9x = 9.0000....
9x = 9
Finally, we divide both sides by 9:
x = 1.

Hope that helps. For more information, see the articles I've linked below.

Answer 5

1/3=.3 recurring
1/3+1/3+1/3=.9 recurring
1/3+1/3+1/3=1
1=.9 recurring

So yes 1 does equal .9 recurring

Answer 6

Wow! Some of the people answering this question need to learn some math before answering a math question!


0.9999.... does indeed equal 1. It has nothing to do with taking limits.

If it doesn't equal 1, then | 1 - 0.999.... | = d > 0 for some constant d.

1.000000000000000000000000 ... 00 ...
- 0.999999999999999999999999 ... 99 ...
________________________________
0.000000000000000000000000 ... 00 ...

Thus every digit of d, after the decimal point, is zero. Thus d = 0.

So 1 = 0.999......

Answer 7

There is actually a way to proove that 0.9(recurring) is equal to one.

One third expressed as a decimal is 0.3(recurring)
Multiply the decimal by 3 and you get 0.9(recurring) which if you multiply the 1/3 by 3 is equal to 1!

Answer 8

It is very frustrating to see the people who are answering you with the answer "no". .9 repeating does indeed equal 1, and there are many ways to prove it.

Some of the people here have called you a 'dingbat' and other names and then they answer the question incorrectly! These kinds of things make me frustrated with Yahoo! answers.

Many people answered your question well including Jay H who provided references to web sites with the proof.

Answer 9

YES, the number 1 has two ways of writing it as a decimal. One is 1.00000..., the other is 0.9999...
Those who say the two are different are simply and utterly wrong. If they are this wrong, I wouldn't trust their answers for anything in math, period.
Every terminating decimal has another representation with recurring 9's. For example,
.25=.24999....

Answer 10

Recurring means that there is no limit to the digits of nine.

That equals one, both in a practical sense and a theoretical sense.